How can I force the Compositor to update via Python?
Compositing output can be updated by rendering without any Render Layers nodes in the compositing node tree. 

See documentation about Render Operators

bpy.ops.render.render(animation=False, write_still=False, use_viewport=False, layer="", scene="")


If I am not mistaken this use_viewport parameter seems to be for controlling if the render process is shown in the UI. It is false by default so it should not show the render in the UI unless specified otherwise.